1. Chapter 5
The Laws of Motion

2. Sir Isaac Newton 1642 – 1727 Formulated basic laws of mechanics
Discovered Law of Universal Gravitation
Invented form of calculus
Many observations dealing with light and optics

3. Force Forces are what cause any change in the velocity of an object
Newton’s definition
A force is that which causes an acceleration

4. Classes of Forces
Contact forces involve physical contact between two objects
Examples a, b, c
Field forces act through empty space
No physical contact is required
Examples d, e, f

5. Fundamental Forces Gravitational force Electromagnetic forces
Between objects
Electromagnetic forces
Between electric charges
Nuclear force
Between subatomic particles
Note: These are all field forces

6. More About Forces
A spring can be used to calibrate the magnitude of a force
Doubling the force causes double the reading on the spring
When both forces are applied, the reading is three times the initial reading

7. Vector Nature of Forces
The forces are applied perpendicularly to each other
The resultant (or net) force is the hypotenuse
Forces are vectors, so you must use the rules for vector addition to find the net force acting on an object

8. Newton’s First Law
If an object does not interact with other objects, it is possible to identify a reference frame in which the object has zero acceleration
This is also called the law of inertia (慣性定律）
It defines a special set of reference frames called inertial frames
We call this an inertial frame of reference

9. Inertial Frames
Any reference frame that moves with constant velocity relative to an inertial frame is itself an inertial frame
A reference frame that moves with constant velocity relative to the distant stars is the best approximation of an inertial frame
We can consider the Earth to be such an inertial frame, although it has a small centripetal acceleration associated with its motion

10. Newton’s First Law – Alternative Statement
In the absence of external forces, when viewed from an inertial reference frame, an object at rest remains at rest and an object in motion continues in motion with a constant velocity
Newton’s First Law describes what happens in the absence of a force
Also tells us that when no force acts on an object, the acceleration of the object is zero

11. Inertia and Mass
The tendency of an object to resist any attempt to change its velocity is called inertia
Mass is that property of an object that specifies how much resistance an object exhibits to changes in its velocity
Masses can be defined in terms of the accelerations produced by a given force acting on them:
The magnitude of the acceleration acting on an object is inversely proportional to its mass

12. More About Mass Mass is an inherent property of an object
Mass is independent of the object’s surroundings
Mass is independent of the method used to measure it
Mass is a scalar quantity
The SI unit of mass is kg

13. Mass vs. Weight Mass and weight are two different quantities
Weight is equal to the magnitude of the gravitational force exerted on the object
Weight will vary with location
Example:
wearth = 180 lb; wmoon ~ 30 lb
mearth = 2 kg; mmoon = 2 kg

14. Newton’s Second Law
When viewed from an inertial reference frame, the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass
Force is the cause of change in motion, as measured by the acceleration
Algebraically,
With a proportionality constant of 1 and speeds much lower than the speed of light

15. More About Newton’s Second Law
is the net force
This is the vector sum of all the forces acting on the object
Newton’s Second Law can be expressed in terms of components:
SFx = m ax
SFy = m ay
SFz = m az

16. Units of Force The SI unit of force is the newton (N)
1 N = 1 kg·m / s2
The US Customary unit of force is a pound (lb)
1 lb = 1 slug·ft / s2
1 N ~ ¼ lb

17. Gravitational Force
The gravitational force, , is the force that the earth exerts on an object
This force is directed toward the center of the earth
From Newton’s Second Law
Its magnitude is called the weight of the object
Weight = Fg= mg

18. More About Weight
Because it is dependent on g, the weight varies with location
g, and therefore the weight, is less at higher altitudes
This can be extended to other planets, but the value of g varies from planet to planet, so the object’s weight will vary from planet to planet
Weight is not an inherent property of the object

19. Gravitational Mass vs. Inertial Mass
In Newton’s Laws, the mass is the inertial mass and measures the resistance to a change in the object’s motion
In the gravitational force, the mass is determining the gravitational attraction between the object and the Earth
Experiments show that gravitational mass and inertial mass have the same value

20. Newton’s Third Law
If two objects interact, the force exerted by object 1 on object 2 is equal in magnitude and opposite in direction to the force exerted by object 2 on object 1
Note on notation: is the force exerted by A on B

21. Newton’s Third Law, Alternative Statements
Forces always occur in pairs
A single isolated force cannot exist
The action force is equal in magnitude to the reaction force and opposite in direction
One of the forces is the action force, the other is the reaction force
It doesn’t matter which is considered the action and which the reaction
The action and reaction forces must act on different objects and be of the same type

22. Action-Reaction Examples, 1
The force exerted by object 1 on object 2 is equal in magnitude and opposite in direction to exerted by object 2 on object 1

23. Action-Reaction Examples, 2
The normal force (table on monitor) is the reaction of the force the monitor exerts on the table
Normal means perpendicular, in this case
The action (Earth on monitor) force is equal in magnitude and opposite in direction to the reaction force, the force the monitor exerts on the Earth

24. Free Body Diagram
In a free body diagram, you want the forces acting on a particular object
Model the object as a particle
The normal force and the force of gravity are the forces that act on the monitor

25. Free Body Diagram, cont.
The most important step in solving problems involving Newton’s Laws is to draw the free body diagram
Be sure to include only the forces acting on the object of interest
Include any field forces acting on the object
Do not assume the normal force equals the weight

26. Applications of Newton’s Law
Assumptions
Objects can be modeled as particles
Interested only in the external forces acting on the object
can neglect reaction forces
Initially dealing with frictionless surfaces
Masses of strings or ropes are negligible
The force the rope exerts is away from the object and parallel to the rope
When a rope attached to an object is pulling it, the magnitude of that force is the tension in the rope

27. Particles in Equilibrium
If the acceleration of an object that can be modeled as a particle is zero, the object is said to be in equilibrium
The model is the particle in equilibrium model
Mathematically, the net force acting on the object is zero

28. Equilibrium, Example 1a
A lamp is suspended from a chain of negligible mass
The forces acting on the lamp are
the downward force of gravity
the upward tension in the chain
Applying equilibrium gives

29. Equilibrium, Example 1b
Not an action-reaction pair
Both act on the lamp
Action-reaction forces
Lamp on chain and chain on lamp
Chain on ceiling and ceiling on chain
Only the forces acting on the lamp are included in the free body diagram

30. Equilibrium, Example 2a Example 5.4 Conceptualize the traffic light
Assume cables don’t break
Nothing is moving
Categorize as an equilibrium problem
No movement, so acceleration is zero
Model as a particle in equilibrium

31. Equilibrium, Example 2b Analyze Need two free-body diagrams
Apply equilibrium equation to the light
Apply equilibrium equations to the knot

32. Equilibrium, Example 2 c Analyze, cont. Finalize
Find T3 from applying equilibrium in the y-direction to the light
Find T1 and T2 from applying equilibrium in the x- and y-directions to the knot
Finalize
Think about different situations and see if the results are reasonable

33. Particles Under a Net Force
If an object that can be modeled as a particle experiences an acceleration, there must be a nonzero net force acting on it
Model is particle under a net force model
Draw a free-body diagram
Apply Newton’s Second Law in component form

34. Newton’s Second Law, Example 1a
Forces acting on the crate:
A tension, acting through the rope, is the magnitude of force
The gravitational force,
The normal force, , exerted by the floor

35. Newton’s Second Law, Example 1b
Apply Newton’s Second Law in component form:
Solve for the unknown(s)
If the tension is constant, then a is constant and the kinematic equations can be used to more fully describe the motion of the crate

36. Note About the Normal Force
The normal force is not always equal to the gravitational force of the object
For example, in this case
may also be less than

37. Inclined Planes Forces acting on the object:
The normal force acts perpendicular to the plane
The gravitational force acts straight down
Choose the coordinate system with x along the incline and y perpendicular to the incline
Replace the force of gravity with its components

38. Multiple Objects
When two or more objects are connected or in contact, Newton’s laws may be applied to the system as a whole and/or to each individual object
Whichever you use to solve the problem, the other approach can be used as a check

39. Multiple Objects, Conceptualize
Observe the two objects in contact
Note the force
Calculate the acceleration
Reverse the direction of the applied force and repeat

40. Multiple Objects, Example 1
First treat the system as a whole:
Apply Newton’s Laws to the individual blocks
Solve for unknown(s)
Check: |P12| = |P21|

41. Multiple Objects, Example 2 – Atwood’s Machine
Forces acting on the objects:
Tension (same for both objects, one string)
Gravitational force
Each object has the same acceleration since they are connected
Draw the free-body diagrams
Apply Newton’s Laws
Solve for the unknown(s)

42. Exploring the Atwood’s Machine
Vary the masses and observe the values of the tension and acceleration
Note the acceleration is the same for both objects
The tension is the same on both sides of the pulley as long as you assume a massless, frictionless pulley

43. Multiple Objects, Example 3
Draw the free-body diagram for each object
One cord, so tension is the same for both objects
Connected, so acceleration is the same for both objects
Apply Newton’s Laws
Solve for the unknown(s)

44. Problem-Solving Hints Newton’s Laws
Conceptualize
Draw a diagram
Choose a convenient coordinate system for each object
Categorize
Is the model a particle in equilibrium?
If so, SF = 0
Is the model a particle under a net force?
If so, SF = m a

45. Problem-Solving Hints Newton’s Laws, cont
Analyze
Draw free-body diagrams for each object
Include only forces acting on the object
Find components along the coordinate axes
Be sure units are consistent
Apply the appropriate equation(s) in component form
Solve for the unknown(s)
Finalize
Check your results for consistency with your free-body diagram
Check extreme values

46. Forces of Friction
When an object is in motion on a surface or through a viscous medium, there will be a resistance to the motion
This is due to the interactions between the object and its environment
This resistance is called the force of friction

47. Forces of Friction, cont.
Friction is proportional to the normal force
ƒs £ µs n and ƒk= µk n
μ is the coefficient of friction
These equations relate the magnitudes of the forces, they are not vector equations
For static friction, the equals sign is valid only at impeding motion, the surfaces are on the verge of slipping
Use the inequality if the surfaces are not on the verge of slipping

48. Forces of Friction, final
The coefficient of friction depends on the surfaces in contact
The force of static friction is generally greater than the force of kinetic friction
The direction of the frictional force is opposite the direction of motion and parallel to the surfaces in contact
The coefficients of friction are nearly independent of the area of contact

49. Static Friction Static friction acts to keep the object from moving
If increases, so does
If decreases, so does
ƒs  µs n
Remember, the equality holds when the surfaces are on the verge of slipping

50. Kinetic Friction
The force of kinetic friction acts when the object is in motion
Although µk can vary with speed, we shall neglect any such variations
ƒk = µk n

51. Explore Forces of Friction
Vary the applied force
Note the value of the frictional force
Compare the values
Note what happens when the can starts to move

52. Some Coefficients of Friction

53. Friction in Newton’s Laws Problems
Friction is a force, so it simply is included in the in Newton’s Laws
The rules of friction allow you to determine the direction and magnitude of the force of friction

54. Friction Example, 1
The block is sliding down the plane, so friction acts up the plane
This setup can be used to experimentally determine the coefficient of friction
µ = tan q (why?)
For µs, use the angle where the block just slips
For µk, use the angle where the block slides down at a constant speed

55. Friction, Example 2
Draw the free-body diagram, including the force of kinetic friction
Opposes the motion
Is parallel to the surfaces in contact
Continue with the solution as with any Newton’s Law problem
This example gives information about the motion which can be used to find the acceleration to use in Newton’s Laws

56. Friction, Example 2 -fk=max and n-mg=0 max = -kn = -kmg ax = -kg
Apply the following equation: vxf2=vxi2+2ax(xf-xi) with xi=0 and vf=0 to determine ax.

57. Friction, Example 3
Friction acts only on the object in contact with another surface
Draw the free-body diagrams
Apply Newton’s Laws as in any other multiple object system problem

58. Friction, Example 3  Fx = Fcos - fk - T = m1ax = m1a
 Fy = n + Fsin - m1g = 0
 Fy = T - m2g = m2a
fk = kn
fk = k(m1g - Fsin)
a = ?

59. Analysis Model Summary
Particle under a net force
If a particle experiences a non-zero net force, its acceleration is related to the force by Newton’s Second Law
May also include using a particle under constant acceleration model to relate force and kinematic information
Particle in equilibrium
If a particle maintains a constant velocity (including a value of zero), the forces on the particle balance and Newton’s Second Law becomes

60. Homework: Take a closer look at examples in the textbook
Homework: Take a closer look at examples in the textbook. (Some of them will appear in the midterm or final exam with changing the parameter.)
Exercise: